Stable-driven SDEs: a Parametrix Approach to Heat Kernel Estimates with an Application to Stochastic Algorithms

Abstract : First, we study a class of stochastic differential equations driven by a possibly tempered Lévy process, under mild conditions on the coefficients (Hölder continuity). We prove the well-posedness of the associated martingale problem as well as the existence of the density of the solution. Two sided heat kernel estimates are given as well. Our approach is based on the Parametrix series expansion. Then, we consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hörmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process. Finally, we obtain an expansion of the implicit weak discretization error for the target of stochastic approximation algorithms introduced and studied in [Fri13]. This allows us to extend and develop the Richardson-Romberg extrapolation method for Monte Carlo linear estimator (introduced in [TT90] and deeply studied in [Pag07]) to the framework of stochastic optimization by means of stochastic approximation algorithms. We notably apply the method to the estimation of the quantile of diffusion processes. Numerical results confirm the theoretical analysis and show a significant reduction in the initial computational cost.
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Submitted on : Tuesday, July 28, 2015 - 11:04:52 AM
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Lorick Huang. Stable-driven SDEs: a Parametrix Approach to Heat Kernel Estimates with an Application to Stochastic Algorithms. Mathematics [math]. Univeristé Paris Diderot Paris 7, 2015. English. ⟨tel-01180708⟩

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